Generalized Rayleigh’s problem in viscous flows

VISCOUS FLOWS1

By J. C. WU and T. YAO-TSU WU

[Received 11 May 1965. Revise 15 August 1966]

SUMMARY

The generalized Rayleigh 's problem is considered as the flow of an incompressible viscous fluid produced by an infinitely long cylindrical solid surface, of arbitrary shape, moving parallel to its generators. A broad review of the literature is in-corporated with further extension to new problems and new methods of solution. Some of the problems previously treated are investigated again with a different, and somewhat simpler approach, yielding the solution in new forms and additional information of interest. For the general caso of an arbitrary cylinder, a perturbation method is established by means of an integral equation which can be solved by iteration.

1. Introduction

RAYLEIGH (1) first noted an exact solution of the Navier-Stokes equations

for an infinite flat plate moving impulsively parallel to itself in a viscous incompressible fluid, in which case the equations reduce to a diffusion equation in one space dimension. When this Rayleigh's problem is generalized to a class of flows due to an infinite cylinder of arbitrary cross-sectional profile moving parallel to its generators in a viscous incompressible fluid, the flow is still uni-directiona], but the velocity depends on both coordinates in the plane normal to the motion.

It

satisfies a two-dimensional diffusion equation, with no convective effect,

so the vorticity generated at the boundary is diffused outwards by

viscous action only. A knowledge of unsteady uni-directional flow of this kind may shed light on the basic features of the more difficult non-linear problem of the steady flow past a semi-infinite body of the same

cross-section. However, as noted by several authors (see (2) to (9)), this generalization also makes the solution considerably more difficult. In fact, in the class of a general cylinder with finite lateral dimensions moving in an infinite fluid, not a single complete solution is known except for the circular cylinder. In the existing literature, the analytical

effort is often found to be content with a result for the asymptotic

¶ This paper was initiated under the Independent Research and Development Program

of the Douglas Aircraft Company, Santa Monica, California, U.S.A.

Professor of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia; Previously with Douglas Aircraft Company.

§ Consultant of Douglas Aircraft Company; Professor of Applied Mechanics, California Institute of Technology, Pasadena, California.

[Quart. Journ. Medi, and Applied Math., Vol. XX, Pt. 4, 1967]

394 J. C. WU ANI) T. YAO-TSU WEJ

behaviour of the local skin friction and the total viscous drag, leaving the detailed flow field virtually unexplored.

The primary purpose of this paper is to examine the different methods of solution which can be applied to the generalized Rayleigh's problem and to carry out the analysis using the most effective method (as seen by the authors) for several specific problems. It is a common experience that the solution obtained by different methods is usually expressed in different forms, each of which may be particularly convenient for a special purpose. Conversion of the solution from one form into another is not always simple. Some of the problems treated here are new, and some of the previously solved problems are investigated again with a different approach which seems to be simpler than the previous attempts,

yielding new forms of solution and some additional information of interest.

In this sense the present paper may be regarded as a review and further development of the generalized Rayleigh's problem.

The scope of the present work may be briefly stated as follows.

First, several methods of solution for the general problem are discussed. The particular method of separation of variables in terms of various similarity forms is further explored, and some simple solutions are obtained for several special cases. This method is applied to treat the flow inside an arbitrary infinite wedge. The asymptotic behaviour of the velocity field and skin friction is examined for small and large values of the time. In particular, Howarth's problem for a semi-infinite plate is

solved by using a Green's function, yielding the solution in an integral representation which is found useful in computation for all values of the time.

The main problem with an arbitrary profile is considered (in section 8) for the entire duration of the impulsive motion. For small values of the time t, Cooke's method (5) is further simplified by applying an integral transform; the skin friction T on a cylinder with no corners is solved to six terms (see eq. (81)) for small vt/R (where r is the kinematic viscosity and E(i) is the local radius of curvature, as a function of the arc length i). The solution shows that the effect of curvature, R, enters in the second-order term, the effect of curvature variation, dR/dl, on the local friction first appears in the fourth-order terms; whereas the effect of curvature variation on the total drag, or the integral of T with respect to i, appears as late as in the fifth-order approximation. For large values of vt/R2, on the other hand, the total drag experienced by an arbitrary cylinder is asymptotically equal, up to a term of any finite power of log (vt/a2), to

that on a related circular cylinder of radius a, where the radius a is

determined by the condition that Idf/dj - i as

-

, z = f() being

an analytic function which represents the conformal transformation of the z-plane outside the arbitrary cylinder into the region

> a. For

intermediate values of vt/R2, a perturbation method is established by means of an integral equation.

Finally, in section 9, the original Rayleigh's analogy between the unsteady motion and the analogous problem of the steady flow past a semi-infinite body of the same cross-section is evaluated for a few specific cases where the corresponding steady solutions for small and large distances from the leading edge are available.

2. Formulation of the generalized Rayleigh's problem

The generalized Rayleigh's problem is referred here to the flow of an incompressible, viscous fluid produced by an iiifinitely long cylindrical solid surface moving parallel to its generators, which are taken to be in the z-direction. The cross-sectional profile, in the (x, y) plane, of the surface is arbitrary and may be written as

S(x, y) = O, or parametrically as x = x(s), y = y(s), (1) this curve being either open or closed, of finite or infinite extent in the (x, y) plane. For the generalized Rayleigh's problem, the only non-vanishing velocity component is w (in the z-direction), which satisfies the equation

dw aw a2w

(2)

where i' is the kinematic viscosity of the fluid and t is the time. The boundary conditions and initial value of w are

w(x, y, t) = f(t) for > O, S(x, y) = O, (3a)

w(x, y, O) = y(x, y) for (x, y) in the flow field, (3b) grad w -* O

for t < a,

infinitely far away from S. (3e)

Condition (3a) states the prescribed motion of the solid surface and the no-slip condition for the velocity at the surface. If the flow is bounded by some closed solid surface, condition (3c) of course does not apply and must be relaxed. As a remark, this fluid-dynamic problem is equivalent to a transient heat conduction problem in which the surface temperature of the cylindrical surface changes with the time described by f(t). In fact, in heat-conduction problems, the wall temperature may be pre-scribed as w = f(x, y, t) onS(x, y) = O, t > O, wheref(x, y, t) is completely arbitrary. This generalization, however, appears to be too artificial for the fluid dynamic interest.

396 J. C. WU AND T. YAO-TSU WIT

Several general methods of solution are available for treating the present linear problem. First, equation (2) admits a solution in which x, y and t appear only in the form x/(vt)& and y/(vt) so that the number of the independent variables is reduced by one. Such a similarity solution may prevail if the initial and boundary conditions can also be expressed in terms of these similarity variables. It is therefore necessary that S(x, y) in (1) must be a homogeneous function of x, y; g(x, y) in (3b) can at most be a constant; andf(t) in (3a) can at most be a piecewise constant function in t (and possibly also in x, y), which corresponds to a step change of the surface velocity. The similarity solution under these

necessary conditions, though of a special form, is nevertheless

of

fundamental importance, for it can be utilized to construct the solution of the general case. If the solution corresponding to the homogeneous initial condition (g(x, y) = O) and a unit step function of the surface velocity (f(t) = 1) is obtained, say F(x, y, t), then for the problem with g(x, y) = O and f(t) arbitrary, the solution follows immediately from the Duhamel integral (10): I aF(x,

y, tT)

w(x, y, t) = f(T) at dT. o

This formula is not restricted to the case of similarity, so that S(x, y) in (1) can be perfectly general. This is the main justification for con-centrating on the impulsive start, as will be done throughout the rest of the paper.

A second method is by making use of the operational calculus, with application of certain integral transforms with respect to t or some space coordinates, or both. The solution by this method generally appears, upon inversion, in an integral representation.

Another method is by construction of a Green's function (or the

fundamental solution) appropriate to the particular boundary

con-figuration in question. The problem then reduces to a surface integration.

For definiteness, we shall define the Green's function G(x, y; ,

; t T) by

(--+

-- -G = ô(x)â(y-7)ô(t--T)

\aX- ay- at1

G(x, y; , ; tT) = O for t <T, (6a)

G(x,y; ,7;tT) = O

onS(,?7) = O

fort >r.

(6b)

Here (x, y) and (, i) are points in the flow field D and 6(x) is the Dirac delta function. Furthermore, G is required to vanish as x2 +y2 -- cc if the point of infinity is a boundary point of D. Then the solution of (2) under

(4)

conditions (3a, b, c) is

w(x, y, t) =

vf dl-Jf(T)ds+jjg(,

)G(x, ; ; t) d (7)

where 8 denotes the boundary curve S(x, y) = O of the domain D of the flow field, and n1 is the inward-drawn normal.

It may be noted that the solution obtained by different methods is usually expressed in different forms. Each may be convenient for some particular purpose, and it is often difficult to deduce one form directly from the other.

3. Some simple solutions by separation of variables

By inspection, w(x, y, t) = X(x, t; A)Y(y, t; A) is a solution of (2) provided

XvX = AX,

Y1vY, = AY,

(Sa)

where 2 is the separation constant and the success of the method depends on the possibility of representing a solution as

w(x, y, t) = aX,(x, t; O)+bY(y, t; O)+cnXn(x, t; A)Y(y, t; A). n

(8b)

A different form of separation of variables may be obtained in terms of the polar coordinates (r, 6), defined by x = r cos O, y = r sin O. In

terms of (r, O) the differential equation (2) becomes

(9) ar2

rar

r2a02 _{' at}

Introducing the new similarity variable

= rJ(4vt), (10)

and considering the problems in which w depends on , O only, (9)

becomes

= O.

_(i')

This equation may further be reduced to the ordinary differential equation d2w dw

+2e = O,

(12a)

d2

d

where

= cos(Oo),

(12b)

o being an arbitrary phase angle (this is implied by the homogeneity of (11) in O) provided that w is in fact a function of only, or w can be

398 J. C. WIJ AND T. YAO-TSU WEJ

constructed, by variation of the parameter , from this form. The

general integral of (12) is

w() =

a0±a1erf(e), (13)

where a,

erf(x) __Jexp(_t2)dt, erfc(x) = 1erf(x). (14)

/1To

Furthermore, it is readily verified that

w(, O) = X(eÍ)X(Ek), (15a)

with

_{= cos(O),}

(15b)

is also a solution of (11) if X() satisfies (12) and if

ck = (n+l)ir, n

being an integer.

It can also be shown that the product of three

functions X()X()X(ek), or with more factors, can no longer satisfy (11) regardless of the choice of , c,., and Therefore, a solution by this

separation of variables may be expressed as

w(, O) = aO+akerf(k)-Hbkerf()erf(?7k), (16a)

where

_{= cos(O),}

71k = sin(Ok).

(16b)

The above summations are taken over arbitrary discrete values of the parameters . These summations may be replaced by an integration

over a continuous variable with ak and bk taken as arbitrary functions

of c; that is

w

c+ff()erf{ cos(O)} dc +Jg()erf{ eos(O)}erf{ sin(O)} d,

or, with integrations by parts,

w = c+J exp{-2cos2(Oo)}{F()-f-G()erf{ siri(Oc)}] sin(O) dOE,

where f, g, F, G are arbitrary functions of c, and c is an arbitrary constant. In this manner the forms (16) to (18) are going to be used to solve the problems of sections 3.1, 3.2 and 3.3.

3.1. Rectangular corner: x > O and y > O

A simple example of the solution assuming the form (16) is the flow in the quadrant x > O, y > O, generated by an impulsive motion of the

planes x = O and y = O with velocity W parallel to themselves. The solution is obviously (see e.g. (10))

w X

=1erf

erf

W .,/(4vt) J(4vt)

On the plate y = 0+, the only nonvanishing component of the stress is x

erf

T =

-(ä1)=0 = (rt)

/(4t)'

which is numerically less than Rayleigh's infinite plate solution by

Ar

-

1uW erfe X (21)

vt) /(4vt)

Thus the presence of a rectangular corner results in a reduction of the total frictional force (from Rayleigh's infinite plate)

co

AD = 2 f (Ar) dx = (4f7i)/zW. (22)

The quantity AD is seen to be independent of the time. 3.2. Infinite wedge, O < O <ir/n, n = 1, 2,...

The flow inside the wedge bounded by planes O = O and O = 7r/n (n being an integer) can be continued periodically into the entire region O <O <2ir by inserting planes atO = miT/n, m

2,3,..., 2n-1. Then

the boundary conditions at these radial plates become

w= W forO=mir/n,

ni=0,l,2,...,2n-1, andt>O;

(23)

W-0 asr,

Ornir/n.

(24) Consequently, w has the symmetry w(0) = w{(iT/n) O} and the

periodic-ity w(0) = w{O +(r/n)}, and hence also the following properties:

w(0) = w{O+(mir/n)} = w{(mir/n,)O}, m = 0, 1, 2...2n-1. (25)

In particular we note that w(0) = w(irO) for all n; and u'(0)

_{= w(.irO)}

holds valid when n is even but not when n is odd. By observing the above properties of w, we obtain by inspection of the solutions (16)_to (18) that in miT/n < O < (m+l)îr/n, m

= 0, 1,..., 2n-1, we have the

400 J. C. WEJ AND T. YAO-TSTJ WIT

w_l i

W 2 s=0

-

sln(O--'I

( )m+ erf{ / S7r k (ii) n = 4N+2, (N = 0, 1, 2,...),

(m<O<m±1)

(

slr\) I I _)m+S

_{erf{ cos O--')}

_{erf smf}

\ i (iii) n = 2(N+l), (N = 0, 1, 2,...). n-1 (8+1)1:/n

=

(_)m+8

f exp-2sin2(Oc)}x

80 sw/fl X erf(oos(Ox)}cos(Om)dcL. (28)

It is rather straightforward to show that these solutions satisfy all the boundary and initial conditions and possess the properties (25). It may

be noted that in the case of n = 4N+2, the integrals in (28) can be

reduced to the form (27). The solution expressed in the above form (with m = O only) has been given by Sowerby (3) by using the method of Green's function. The present method is considerably simpler.

The skin friction on the plate O = O is 'r n-1 ( ST ( 21S7T\ ) 1iW

()S

)expsin (_)

(n2N+1),

(29) \/(ITVt)

n,

\n/)

- (/r) ( aw/aO or (26) (27) Sir\ ( . /sir\ ( /s'n

(_)scos()explsmn(I erf) cos_)

, (n = 2N).

\Th/ \Th/) k

n,J

(30)

This result shows that the skin friction deviates from Rayleigh's

solution by an exponentially small term for large . The other properties

of the flow will be discussed together with the general case of arbitrary

wedge angle (see section 4).

3.3. Flow within the rectangle -a <x <a, -b <y <b.

Another example of the method of separation of variables is the flow

inside a rectangle bounded by the walls at x = +a and at y =

which are set into an impulsive uniform motion f(t) = WH(t) in the z-direction, H(t) being the Heaviside step function. The solution is

known (see, e.g. 10, section 5.6); we merely recite jt below for subsequent following oases: (i) n = 2N+1, (N = 0, 1, 2,.), n w W _s=0

r=

./(.rrvt)

comparison regarding the corner effect.

w/W = 1(x, t; a)tp(y, t; b), (31) where

(x, t; a) = cos (2n+1)irx _expl ir2vt(2n+1)2 . (32)

ir_{,, =02n+1} 2a 4a2

Another expression for i' in the form (16) can be written as

(x, t; a) = 1

(__)[erfc{2

}+ erfc2

±X}], (33)

which is effectively the solution found by the method of images for the

strip a < x < a.

The series in (33) converges quite rapidly except for large values of vt/a2, in which case the convergence of the series in

(32) is fast.

The total frictional force due to the interior flow is D =

_2JT(x

b)

_{dx+2Jr(_a}

y) dy.

By using the above solutions (32) and (33), we readily deduce that

D

_{4(a+b) l6+}

/ _{a e_b2/,} b

W (irvt) ir (J(vt) (vt) j

D

64b a

ir2vtl

1

4

(-+,

(35)

where (34) is valid for small vt/a2 and vt/b2, and (35) for large. vt/a2 and vt/b2.

Therefore, for small values of vt/a2 and vt/b2, the total frictional force

is less than for the flat plate value (of equal circumference) by, from (34),

4(

+o(

a b _{_a2t\.}

(36)

/2W

\irj

\./(vt) ,J(vt) _J

This force reduction LD clearly comes from the correction due to four

right corners (see (22)), the error of this approximation being exponentially

small. On the other hand, for large 'vt/a2 and vt/b2, the total frictional drag decays exponentially with increasing t.

4. The flow inside an arbitrary wedge O <O <x <21r

The problem of Rayleigh's flow in an arbitrary wedge bounded by O = O and O = oc, O <c <2ir, has been treated by Hasimoto (6); the

velocity field of this flow, however, has not been evaluated.

The corresponding heat conduction problem hs been previously solved by

402 J. C. WtJ AND T. YAO-TSU WIJ

using a Green's function or by the Laplace transform, as can be found in Carsiaw and Jaeger (10, section 14.14, section 15.11). We present below

a different, and somewhat simpler, derivation of the velocity field and its asymptotic behaviour. The differential equation in terms of (, 0) is given by (11); the initial and boundary conditions are

w(, 0)

=

w(, a)

=

W, (0

<

), (37a)

w-0 as ¿-

a,

(0<0 <oc). (37b)

Here the original initial condition at t

=

O and the boundary condition at r

=

a (for t > 0) combine into a single condition (37b). Now from the

symmetry we require that w(, 0) = w(, osO).

Hence we seek a

solution of the form

w/W

=

1+

f()sins0,

s

=

(2n+1)r/,

(38)

where, by (Il), f, satisfies

= 0. (39) The solution (38) satisfies (37a). The remaining conditions at O

and require that f(0)

=

O and f()

=

4{(2+l)ir}' since

sins0=,

(0<O<oc).

02m+1 4

We next introduce the Hankel transform of f with respect to the Bessel function J9 by

g(2)

=

f()J3(Ag d.

Multiplying (39) by J(A) and integrating, we find

' 2'

d.t

(+)u =

which has the solution

g(2)

=

C2exp(-2),

C being an integration constant. Hence by the inverse transform,

f)

=

f g(A)J) dA

=

C f e(A2)J3(A) dA/A,

0

The above integral may also be expressed in terms of the hypergeometric function (see, e.g. 11, section 13.4), giving

or

T

/LW '''F(s)

= ci\/(vt) ' F(s)

1F1(js; 8+1;

2)

n=0

in which C is determined by the condition of f at

= , giving

du

- ü um f exp {(u/2)2}J,(u) =

Cf

J,(v)

=

2n-+-1

'-

u u 8

o o

or = 4/oc. Therefore we obtain the solution as

w 4 . r dA n=0

= 1

sin se _s+1;

_2),

W

c-'-

_F(s+1) or )mF(m+is) sO,

= 1

F(m+s+1) or

s.

_{orn sO}

f exp (_zt){t(1_t)}.

(40d) W

c-F(s+1)

_n-0 t o

Here (40e) follows from the series expansion, and (40d) from an integral representation of 1E1(a; e; z) (see e.g. (12)).

It is readily shown that

(40a) reduces to Rayleigh's solution for the infinite plate with s = 2n. +1 on account of (see e.g. 11, section 2.2.2)

2 J1(z) sin (2n+1)8 = sin (z sin O).

n=0

For a semi-infinite plate,

= 2i, s = n+, (40b) is identical to the

first form of Howarth's solution (2). However, even for the particular case of = IT/n (n being an integer), reduction of the preceding expres-sions to the simple form (26), (27), (28), and in particular the form (19) for n = 2, still involves considerable manipulation.

4.1. Skin friction and total drag

The skin friction on the wall O = 0, r = u/r)(aw/aO)0, is

T

>f exp

(_A2)J5(2A& (41a)

(41b) confluent

404 J. C. WtJ AND T. YAO-TSU WU T -' 1AW ' 'c ()msF(m+s)2m+,_i ¡(vi) L _{in! F(m+s+1)} fl.07fl0

8l

21uW J'

exp (_2t)t(1t)

nO _o

The above expressions result from termwise differentiation, as can be

justified by the uniform convergence of the resulting series.

For arbitrary value of s (2n+1)ir/a, the asymptotic behaviour of T near

the vertex r

=

O can be readily computed from (41e). Of particular importance we note that

/LW

_{F(ir/2)c/_1{i+ F{(irf2)+J (2)2_}

_{+O(48, 4)}}

t.,/(vt) F(ir/c) _{F[(3ir/2ci)+kJ}

2(ir+)

(42) as

=

r(4vt)

-

0. Therefore T tends to zero with r (for finite t) if c < ir,

and T becomes singular like r' as r

O when > ir. The strongest

singularity is at

=

21T, in which case r is of order r4 as r-+0, as has been pointed out by llowarth (2). The above asymptotic behaviour of T

in the range 000i <

< 1 has been calculated by using (42) for several values of , the result being given in Fig. 1.

On the other hand, for moderate and large values of , the value of r can be evaluated as follows. First, the series in (41d) can be summed by making use of the Hankel contour-integral representation of 1/F(z)

(12, Vol. 1, p. 13): (0+) t8-1 1 j' to -

=- I

dv

_(treal,>O), (43) F(z)

2iiJ

y8 B

where Br denotes a contour circumventing the entire negative real

v-axis in the counterclockwise sense, giving

r

2W i

exp (-2t)(1t)

dtj exp {2t(1_t)v}dV

x/(vt) 2iri J

v_v'2

o (J

in which the contour G envelopes the negative real v-axis and the circle

=

1 in the counter-clockwise sense since the convergence of the series

requires rvr > 1.

Now the integrand of the v-integral has branch

points at y

=

O, when ir/2z is not an integer, and a number of simple poles at y

=

n

=

0, 1, 2,. . .,N, N < J2 _{< N+ 1.} (44) or T

=

or T

=

I0

s 6

aô

¿ óÓ r

Distance from the edge of the wedge 2(vty

Fig. i

When 7r/2 is not an integer, we introduce a branch cut along the entire negative real v-axis, there being 2N+l simple poles on the unit circle Ivi = i inside the contour C on the cut plane. When /2r = N + 1, an integer, the contour C reduces to one enclosing the unit circle ivi = 1, containing 2(N+1) simple poles, with an additional pole at y

= 1.

Both these two cases can be treated together. By applying the theorem of residues and using the relation

i

ßI(ß)

₌

_ßJ

exp {t+ß2t(lt)}(1t) dt

o

(ei)+

l+ßexp

f2(1

ß)21\/lrrerf ((1+ß)

_erf

22

_2Jß

_{4ß J 2 L}

_2/ß

J

2Jß

J] * o (45)

III

°° 180°

IIIL

-

165°

iiiuiuumur

08 06 04 02 008 006 004 t) 02

408 J. C. W[J AxO T. YAO-TSLJ WIJ

where ß is an arbitrary complex constant, we finally obtain

IILW T/

vt) (_)" cos(nc) exp

{_2 sin2 (nc)}erf(1 cos n)

+

sin(yir)eJ cosh(yu)sinh u 1F'( sinh iu)du (46a)

IT cosli 2yucos 2yir

o

where y = ii/2z, and

W(x) = exp(_x2)Je 9)2 d9). _(46b)

We note that the above general result reduces to (30) for the special case y = N, an integer, since the second term with sin ylT vanishes where y = N. The asymptotic behaviour of r for large e can readily be deduced from (46a) and the asymptotic expansion of T(x).

The difference iìD between the total skin friction over the wedge surface and that over the infinite plate can be derived by integrating

(41d),

LD = 2J{r(r, t; )r(r, t; 7r)}dr

4W

fL:(1_t_(1_t1.

n=o Summing the series, we obtain

4 1/ 2yx x

\ dx

'ir

-I

, y>

(4e)

1uW

r.) \1x2 1xJlx

2

It is easy to verify that (I)(2ir) =

,

I(ir) = 0, Ir/2) = 2/IT, and

= - co. For other values of x, t'(c) can be readily calculated by numerical integration of the above integral. A different expression for «(os) has been given by Hasimoto (6).

4.2. TIte velocityfield

For small values of

= r(4vt), (40c) is already in the form of the

required asymptotic expansion and is convenient for computation for

O < < 1.

For moderate and large values of , w canbest be evaluated by making use of (43) in (40d), and summing the series, giving

r Re

Re i dv l d

= 1

_j

exp (_2t)_

fexp

_{{t(l_t)v}}

[R2 2i

R2_e2] y'

W 2IT o C

where

R = y7,

=

= rO/,

(0 _<

and the contour C circumvents the negative real v-axis and the circle Ivi = i in the counter-clockwise sense. The integrand again has branch points at y = O, when y is not an integer, for which case a branch cut is made along the entire negative real v-axis. Furthermore the integrand has a number of simple poles in the cut plane at

y =

_{n = N,..., 1,0, 1, 2,..., N, N <}

_N1.

By deforming the contour C to small circles around these poles and a

contour circumventing the entire negative real axis, applying the

theorem of residues to the integration around the small circles, we finally obtain

w_l z

w

(_)fl

_{f exp{-2t+2t(1 t)cos 2(O+n)} x}

x sin{2t(1t)sin

2(O+n)}+1 J exp(_2t)J {exp-2t(1t)t}

x

I

(u7u)cos(yir+c)

(u7u)eos(y7rcr)

du ₍₄₈₎

ku27+u_27_2 cos 2(yir±r) u2+u2-2 cos 2(yiroJ u

The above integral representation of w is convenient for computation

when t is moderate and large. In fact, the usual methods of asymptotic expansion by using Watson's lemma and the method of steepest descent can readily be applied to these integrals for large .

5. Howarth's problem for a semi-infinite plate

From the historical point of view, the original Rayleigh's problem was

first extended by Howarth (2) to the case of a semi-infinite plate.

Though this special ease of c = 2ir is already included in the general case of an arbitrary wedge treated in the previous section, the solution

in a relatively simple form for providing a detailed knowledge of the flow

field is still of particular value. Howarth has obtained the solution in two forms: (i) a solution in series of the equations in polar coordinates, (ii) a solution by operational methods of the equations in parabolic coordinates. The first series converges rapidly for = r(4vt)

< i but

the convergence is slower for large values of . On the other hand, the

operational solution is suitable for computation for > i but is not ideal

for

i < i.

Consequently, it is desirable to develop the solution in

another form which is suitable for computation for all values of .

408 J. C. WU AND T. YAO-TSU WtJ

In this section the method of Green's function is adopted to obtain a solution in the form of a single integral which is ideal for all com-putational purposes. The simplicity of this solution facilitates investi-gation of some additional features of the flow field.

The Green's function for the semi-infinite plate problem can be

derived, from the general expression given by Carslaw (24; see also 9 p. 380, eqs. (9, 10)) for heat conduction in a wedge, by setting = 2IT,

and the result is

G(r, O; r1, O ; T) [exp(_R2/4vT){1+erf{(rrlfvT)cos (0O)}}

Sirvr

exp( R/4vT){1 +erf{(rri/vr) cos (O+O)}}], (49a) where

R2 = r2+r-2rr1 cos(0-01), (49b)

R = r2+r-2rr1 cos(O+O1). (49c)

By using the general formula (7), we obtain t

1riaG

IaG\ w(r, O, t)

-vf dT J dr1 w o o r sin 0 d (r+r1)2

=

Ç_Çe

4VT J 477V T o o where p(r, O; r1) = (rrjIvi-)cos2O.

The integration with respect to T can be readily carried out (by integra-tion by parts, using the new variable o = t/T), giving, for O <O < 277,

dp (50a) W Sifl O erfe A+ exp(B2_A2)erfB}A22, W o where

A = (l+p)

B = 2pcos -0,

= r(4vt).

(50b)

Or, by successive transformations of variables

= (1--z--z,

z = sin(0)tan , (51)

we obtain _,r12

W

i

" Ierfc A+ exp(B2_A2)erf B) d, (52)

A

The above integral representation (52) of the solution is ideal for numerical computation for all values of O < 2; < co, O < O <27T since

the integrand is everywhere bounded. It is immediately seen from (52) that w

=

W at O

=

O and 2ir since as O

-

O, 2ir, both A and B tend to

22; and erf A +erfc A

=

1.

6. Problems for a circular cylinder, r = a

It has been noted that the circular cylinder is probably the only case in the class of a general cylinder with finite lateral dimensions for which the problem has a known complete solution. It is therefore desirable to summarize the important results here as a reference for comparison with the general case of arbitrary cylinders.

When an infinitely long circular cylindrical shell of radius r

=

a, imbedded in an incompressible viscous fluid, is set impulsively in motion

with a constant velocity W parallel to its axis, the axial velocity w has the following integral representation (see e.g. 10, Chapter 13)

w(r, t)

=

2J

est{Ko(r/)/Ko(a\/)}!

(r> a),

(53a)

2if

e9t{i(r\/8)/i(a\/8)}d8

(r <a), (53b)

where C is the Bromwich contour, from s

=

b i co to s

=

b +i cc, b

being real and positive, and I,,, K denote the modified Bessel functions of the first and second kinds.

The skin friction on the outer and inner sides of the cylinder has the following integral representation

=

=

1eíK1(a

Ii

/K0(a ¡

, (54a)

\ar;r=a+

2irij

_\

\vJ/

_\

v/ /(vs)

()ra

=

Je{ii (a/) /i (ap)

}.

(54h) For small values of T = vt/a2, _{the asymptotic behaviour of r can be}

obtained by taking the asymptotic expansion of the integrand for large s. By using the following expansions which are valid for large values of

z,

arg z! < 1712, K(z)

=

(fe_z{i

+

(J))(8

+(4v2:v;32) +0 ()),

(55a) / i eZ

4v-1

(4v2 i)(4v2 32) 1(z)

(2z)

_(1!)(8z)+ (2!)(8z)2 5O.4 Ee (55b)

where where (58) da

j d

i A _. e'1(log o)' =

2a j

a (dz) _F(1z)Jz=o' (59b) C.

the last expression can be derived with the aid of Hankel's contour integral representation of 1/F (cf. (43)). In particular,

A0 = 1,

A1 = -y.

A2 = 2y2_r2/6. (59e)

The above result exhibits several significant features of the skin-friction due to the effect of curvature of the solid surface transverse to the flow direction. First, (56) clearly indicates that for small values of

410 J. C. WIT AND T. YAO-TSIJ WIT

one readily obtains that for T vt/a2 1,

7± e,

t){1+T)4±T!TL:T+o(Ti)}.

(56)

For large values of T vt/a2, the asymptotic expansion for can be obtained by deforming the contour C in (54b) to one enveloping the entire negative real s-axis, then it is readily shown that

r.. = (2iW/a)

exp(vt/a2),

(57a)

fl i

where ,, are the zeros of J0(z). The asymptotic representation of T for large T is therefore

(2,uW/a)exp(rT), = 24048. (57b)

The corresponding value of T

can be obtained with the aid of the

expansion of K0(z) for small z (and K1(z) =

/z\'12

K0(z) =

-

log(C,z), n=0 Cn

e'''

(z) = 1\

(1) = y.

(n+1) = _+(1+++...+_)

(n = 1,2,...), n

and y = 05772, the Euler constant. Thus

/1W

f

ezt{jog( C0./z) }-'[1 + O(z log z)]

±

-2 Tria

L

2/1W

_{'y (_)A(log(C/4T)}">[1 +O(T'log T)],}

_(59a) a _n=0

T (i.e., with the 'diffusion length' ,,/(vt) small compared with the radius of curvature a), the effect of surface curvature is not significant, the resultant skin-friction differing from Rayleigh's flat plate value by

=

T±

2S(+o(T2)

(60)

2a 2'yra/ 4a

48ir \aJ

If we denote the uniform curvature of the cylindrical surface by 1/1?

and assign it positive (or negative) when the surface is

convex (or

concave) towards the transverse flow, the above expression can be combined to give W(

i /vt\

vt T =

TRT, - i ---

I I

+

2R ZR\ir) 4W 25 (vt)1 48 r'R3 \R2) (61)

It is of interest to note that the first-order term of tT for small T is

proportional to the curvature 1/B and is independent of the time t

(more precisely, it is a Heaviside step function). On physical grounds it may be expected that this curvature effect at small time must be a local

behaviour over a body of non-uniform curvature since such effect

contributed by the points at large distances cannot influence thenear region on account of the small diffusion length.

On the other hand, the effect of curvature becomes predominant for large values of T, as can be seen from (57) and (59). In this time interval T+ decays logarithmically slowly and T_ exponentially fast compared with the square-root decay ofToeS More precisely,

(4irT)/log(4T/C)-+ cc _{as T - cc,}

T_/Toe -(4rT)1exp(oT)-0

asT-+

co. (62)

Furthermore, (59) includes a limiting case of particular interest. For fixed vt, one obtains, by letting the radius a become vanishingly small,

7.+ I1W/{a1ogy0a}, (63)

valid for a vt. Thus the skin-friction over such an infinitesimal wire becomes singular as a -+ 0. The total skin-friction D = 2lraT+,however,

vanishes in the limit as a

- 0.

As another point of interest, let us compare the total skin-friction on

the inner side of a circular duct and that of

a square duct of equal surface area, that is, with the square duct spacing c

=

ira/2, a being the radius of the circular duct. The total skin-friction of the _{square duct is} readily deduced by setting a

=

b = c/2 in (34), (35), and for the circular duct we have D

=

2raT_, where T is given by (56) and (57). Thus we

412 J. C. WtJ T. YAO-TSU WU

obtain

1+'\(+o(1

D8 2 (vt .a2),

(64a)

exp{(c-8)vt/a2}

0.97 exp(2217vt/a2) (vt a2). (64b) It therefore follows that on the basis of equal circumferential area, the total skin-friction D8 of the circular duct is slightly greater than that of the square duct for small vt/a2, and DCIDS becomes exponentially large

for large time.

7. Rotating circular cylinder

It is also of interest to note that for the case of infinite circular cylinder,

the flow motions due respectively to the pure rotation (about the axis) and axial translation of the surface are not coupled and hence can be linearly superimposed. Furthermore, the velocity components of these two modes are governed by linear differential equations. The rotational

motion in this particular case may thus be regarded as a further

generalization of Rayleigh's problem. In order to investigate the effect

of curvature in the flow direction, we include here for our later discussion

a brief presentation of the rotational motion. It may be remarked that this problem does not have a corresponding heat-conduction counterpart. This problem has also been treated by Mallick (14).

In this rotational flow the tangential velocity component y and the pressure p satisfy the equations

av

_{= vt- (rv) ,}

alla

₍₆₅₎ at ar,rar

(66) r

with the initial and boundary conditions

v(r, 0, 0) = O for all rand 0, (67a)

v(a, 0, t) = V for

t> 0.

(67b)

Initially p may be taken everywhere constant, a value which p assumes

at r = cc for all t.

With an application of the Laplace transform one obtains v(r,t) = _!_

J

e8{Ki(r,/!) /Ki(aJ)

(r > a), (68a)

in obvious notations. The corresponding skin-friction is given by

T = -(-

=

i2+-__

f

eK0 dzì1 (69a)

\T

rJra

a

2ri

X1(..Jz),,/z)

=

IL

=

-2+

1 ç _(69b) ar rjr_-a- a 2m _J 11(,Jz) /z C

where T

=

vt/a2. The asymptotic behaviour of T can again be derived by similar methods as used in the previous section. By making use of the expansions (55) one obtains, for small values of T,

'

(70)

a (7rT) 2 4\IT; 8 32mr

For large values of T, the contour integral in (69a) can be evaluated by making use of the expansion of K1(z) for small z (which can be derived from (58)) and using known inversion integrals (13, Vol. I, p. 283),

giving for T 1,

r '_. {2+'J eZTK (\/z) dz} 2dUV{l+le_1/4T} (71a)

For the integral in (69b) it is convenient to deform the contour C to one

enveloping the negative real z-axis, giving

exp( ßT),

J1(ß)

=

0,

_{ß =}

38317, (71b)

fl=1

which is already in the form of an asymptotic expansion for large T.

The effect of curvature in the streamwise direction can again be

measured by the deviation of r from Rayleigh's solution. For small time, this difference is, by (70),

=

3ILV111( it\i(t\

7 (3(2

2 a

2yra/

_{4a j}

l6mr \7

\aj

Comparison of this result with (60) shows that for small time, the differ-ence tr due to the streamwise curvature is three times as large as that due to an equal transverse curvature. Equation (7 la) shows that as t

+

cc, i- tends to the steady state limit 2,uVJa, the transient term being of order (a2/i't). This steady state solution of_{T+ corresponds to the} steady velocity field y

=

Va/r (a potential flow) for r > a, which is the limiting solution of (68a) as t

-

cc. _{Finally, the skin friction r_ on} the inner side decays exponentially for large time,as shown by (7 lb). By

414 J. C. 1VU AND T. YAO-TSIJ WEJ

comparing (7 la) with (59) we see that for large values of the time the flow on the convex side due to the effect of the streamwise curvature is distinctively different from that under the influence of a transverse

curvature. The flow on the concave side of a surface decays exponentially

for large time whether the surface curvature is streamwise or transverse; the attenuation is faster in the case of streamwise curvature, as clearly indicated by (57) and (7 lb).

8. Cylinder of arbitrary shape

The formulation of the general Rayleigh's problem for a cylinder of arbitrary shape has beeii given previously in section 2. The first attempt at a solution to this problem was made by Batchelor (4), who succeeded in giving the first two terms of the asymptotic expansion for small time

t and the first term for large t.

The special case of bodies having a number of corners has also beenincludedin the discussion. Subsequently, the asymptotic solution for large t has been extended by Hasimoto (7) by using the Laplace transform and conformal mapping techniques, yielding the result correct to the order of a2/vt, where a is a characteristic length of the cross section. In particular, when a2fmt is sufficiently small, the total frictional drag is shown to depend only on atm/vt, irrespective of the cross-sectional form. In a subsequent paper Hasimoto (8) applied the WKB method for the case of small time t, giving the asymptotic expansion of the skin-friction to four terms. An interesting feature of the solution is that the effect of variation in the curvature of the solid surface on the local skin-friction first appears in the fourth-order term. This asymptotic expansion for small t has been further extended to five terms by Cooke (5) by a different and more straightforward method of expansion. One advantage of Cooke's method is that the velocity field can be readily obtained, whereas Hasimoto's method is not as convenient for this purpose.

In the following Cooke's method is first described with further

simplification by using the Laplace transform. Then a new method is developed, based on the idea of a perturbation theory with respect to a basic flow, in which the solid surface may be transformed conformally into an appropriately chosen simple form, such as an equivalent circular cylinder. This method yields an integral equation which can generally be solved by iteration for arbitrary t. When t is either small or large,

however, the effort in solving the integral equation can be largely curtailed

by using expansion forms of the solution. This method is particularly simple for large t.

8.1. Small time expansion

A straightforward method for the case of small t is by choosing an appropriate set of new variables so that an expansion of the solution may be expressed in terms of these variables. Following Cooke (5), we take a curvilinear coordinate system (A, )

in the (x, y) plane such that A

measures the distance along the normal to the cylinder with the

cylindrical surface given by 2 = O, and

is taken to be the angle

between a plane which passes through the normal to the surface (and is parallel to the generators) and some fixed plane belonging to the family. We further take p = R + A where R() is the radius of curvature of the curve A = O (R being a function of only) and A is positive if it points away from the centre of curvature. Then for a line element of length ds,

(ds)2 = (clA)2+p2(dc142 = (dA)2+(R+A)2(dçh)2.

Strictly speaking, the coordinate system (A, ç) is not necessarily a one-one

correspondence with the points of the plane unless the plane curve of the body surface has no points of inflection. Furthermore, (A,

) are not

defined at sharp corners of the plane curve, if any. However, even in these cases the present method may be useful either with a separate consideration of the local solution and a matching process, or with first

removal of the points of inflexion and sharp corners by a suitable

conformal transformation before this method is applied.

In terms of (A, ) the basic equation (2), after taking the Laplace transform, as defined by

(A, , s) = Jetw(A, , t) di, (73)

becomes

i I

i a ii

?2\ s

- I p +---1 ---I--w = 0,

(74)

paA\ aAj papaj

with the conditions t' = W/s at A = O and i, = O at A =

Next we introduce the variables, as suggested by the similarity solution,

p = R+A = R(1+ii), (75)

and assume the expansion

= (W/s)

₎

(76)

nU

The above form of expansion is clearly suggested by the circular cylinder result. Substituting (76) in (74) and equating coefficients of powers of

i V

416 J. C. WU AND T. YAO-TSU WtY we obtain

n(Pn

,))(n_rn_l)a1Prn+ n=o n1 m=O n-2

+

r

(n_m_1)(_?1) m-2) n=2 ,n=O

ra2_bn+3e+m+mb2 db\

_{= O,} (77a) 2 a 2 dç6) where

b = R' dR/d.

(77b)

The boundary conditions are

ip0(O,

) = i,

ç6) = O, (n = 1, 2,...);

Pn('

ç6) = O, (n = 0, 1,2,...). (78)

Now, by setting the a2

coefficients of ¿" in (77) to zero, we have

-

' a?1 (79a) (79b)

=

o

-o

- )

(n-m1)aPm m=O )(n_m2) (n

rni)

_X rn=O

X[a2?p,fl_bn+3rn am+ (mn+mb2 db\ , 1. (79e)

2 ç6 2 dç6) _tmJ

(79e) being valid for n = 2, 3, 4,.... From this set of equations and the boundary conditions (78) (in which the dependence on the angle ç6 does not appear), it is readily seen that , are independent of the angle

ç6, and the effect of curvature variation (b and dbfdç6, which is a

geo-metrical effect) first appears in . Moreover, the effect of circumferential

variation of the flow field (a?pm/ a9, a dynamical effect) arises through the

term D5 since a2/aç6 first appears in the equation for . Since the

variation ?pTh with respect to ç6 is always of higher order than , and its n-derivatives, this set of equations (79) are actually ordinary differential equations. They are readily integrated, and the required solutions of the

first five terms are found to be

= e',

= 1)e",

V2 = 1P3 =

_(5,73+32 +2)e+ (_2b2) (i2+)e',

(80b) P4 =

₊

128 b2

ldh_3

----(

+8,f+7i)e,

(80c) )4(2O17+33?)+3Oi7)l6_dçf.

i/io = __ _(635+7O174+1O5o73+i29,72± 1O4,)e 256 b° l3Oof+ 189,72+ 129,7)e+

+j-

(11,74+22,73+28,72+14,7)e_+ 64 dç6

+(,73+3o72+3))e{ao_7bä+3 (4b2_)

}

(_2b2).

(SOd) Finally, the velocity field is determined by applying the inverse Laplace transform to (76). The final expression for w(x, y, t), however, will be omitted here since this process is rather straightforward.

The skin-friction r has the following Laplace transform

i (aû\

i /(s\(&th\

i

uW Wa2))=O W \j

v)1,)q=o

%/(V i db 25 7db

=

(vs)[1e+(1+b

_--(+5v

+

-

4 32 +3Ç{13+43b2_7

+2(-7b-

+l2b2_3)

( _2b2)}+...]. d

d2

d d d

Application of the inverse transform finally yields

r 1(vt\t

o+1)t(S+)

(vt) _{25 4}

= (7rvtY4+K-- Ì K

-

-

[ +ïKK"+(K')2]+ jAW

4ir)

Ir

+

(2[13K5 (80a) (81)

where K = 1/R is the local curvature, the primes denote differentiations with respect to i, ic' = dic/dl, etc., i is the circumferential arc length on the cylinder, dl = R(ç) dç. In this asymptotic expansion ofT for small t,

415 J. C. WU AND T. YAO-TSU WO

terms is derived by Hasimoto (8) using the WKB method. In a later paper by Cooke (5), the expansion is carried out to five terms, up to the order (vt), his last term, however, differs from (81) in that the term

(K')2 is missing in the term of O(vt).

To the first-order approximation, the total friction per unit length on a cylinder with total arc length L and Ncorners of angle ci,, (n

=

1, 2,..., N)

is found to be _N

D/W =

L/(1TVt) + *I(OC), (81a) where 'LI(o) is given by (47). This formula was first observed by Batchelor

(4) (aside from a missing factor in Batchelor's expression, as was later pointed out by Hasimoto (9)).

8.2. A perturbation method for moderate and large time

It may be noted (e.g. from the circular cylinder problem) that the

steady state limit of the solution (i.e. the limit of w as t - cc) is not

uniformly valid at the point of infinity. In order to avoid this difficulty, we introduce the formulation for moderate and large time consisting of the following steps: (1) conformal transformation of the solid boundary (with closed contour) into a circular cylinder, (2) introduction of the reference flow past the related circular cylinder, and (3) the original problem formulated as a perturbation of the reference flow. We intro-duce the complex variables z

x+iy and

= +i so that the region

outside the solid boundary S(x, y)

=

O in the original z-plane is mapped conformally onto the exterior of the circle of radius a, whose value is to be determined, in the c-plane by an analytic function

z =f().

(82)

It may bc remarked that the mapping of the interior of S(x, y)

=

O onto

the interior of the circle ¿

=

a2 is generally by means of a different analytical function, say z

= F().

For the general discussion, however, we may usef() to denote both the exterior and interior flows. In terms of the t-plane, the governing equation (2) becomes

a2v

= --i-i--

4df /df\ w

(t>O, > a or

<a),

(83)

w being now considered as a function of , its complex conjugate , and t.

With w kept invariant under the mapping, the boundary and initial values are

_w(,

-,t) = WH(t) on

=

a, (b4a)

w(,

, O)

=

O for J a, (84b)

Condition (84c) is, of course, not required for the interior problem. It may be noted that the simplification of the original geometry to a circle

is achieved at the expense of working with a more complicated differential

equation (83), which now has a variable coefficient. However, from the circular cylinder problem it may also be noted that the resulting flow at small time instants is of a surface layer phenomenon, and in the large time limit, the flow at large distances can recognize only the total drag of the cylinder, and is insensitive to the detailed local geometry.

Based on this point of view, we first write (83) as

a2a2

i'at)v'

'at'

(85)

where

_f'

_{denotes df/d and c is a real constant to be so chosen that a}

perturbation method can be suitably established. We next decompose w into two parts

w(E, , t)

=

w0(, ,, t)+v(, , t), (86) with w0 defined by

'a2 a2 2a'

= o

a2 a2

aty

and w0 satisfies the same boundary and initial conditions (84) as forw

so that u'0 is the solution of a related circular cylinder problem in the t-plane, which is known (aside from the parameter ), as given in the previous section. The remaining problem for y is thus prescribed by

'(ff-2)a(WO+V)

(lI

a), (88a)

v(e, i, t)

=

O on

=

a, t> O, (88b)

v(,

, O)

=

O for all , , (88c)

v(, ,j, t)

*

O as --- . (88d)

Thus y satisfies an inhomogeneous differential equation with homogeneous

boundary and initial conditions. The inhomogeneous term of the differ-ential equation is due to the basic flow w0. This differdiffer-ential equation can be converted to an integral equation as follows.

By applying the Laplace transform (73), (88a) is transformed under the homogeneous initial condition to

\aaJ

=

8(ff

2)(+)

_{= g(, ,, s).}

₍₈₉₎

TI _{Green's function , ; ¿', ';s) may be defined by the equation}

a220

₌

a'-

a7'-

_/

(t>O,lJa),

(87)

420 J. C. WU AD T. YAO-TSU W[J

with r2

= 2+2

> a2,

r'2 = '2+'2

> a for the exterior flow, or with

r <a, r' < a for the interior flow,

satisfying the conditions (E ; ', ,'; s)

=

O on r'

=

a,

Û O as r' co, (90b)

Ûis everywhere bounded except at

=

', ,j =

The solution is readily shown to possess the singular part

; ', ,j';s)

=

(1/2r)K0(kR),

where k

=

/(s/v), R2 = r2+r'2_2rr' cos (Ofi'), (r, O), and (r', O') being respectively the polar coordinates corresponding to (,

) and

(E' j'). The above modified Bessel function K0(kR) may further be

expanded by the addition theorem (11) as

K0(kR)

=

n

I(kr)K(kr')cos n(OO') (r' > r),

=I(kr')K,,(kr)cos 'n(OO')

(r' <r).

Let G

₌

O + ; the regular part may assume the following expansion

=

A,K(kr')cos n(OO') (r' > a),

n

_B,LIl(kr')cos n(OO')

(0 r' < a),

where the coefficients A and B can be determined by condition (90b). From this it immediately follows that for the exterior flow, r' > a,

F(kr',

ka)K')

cos (OO')

(a

r' <r),

(91) K(1ca)

where

F(x, y)

=

I(x)K(y)K,(x)I(y),

(92)

while for a < r < r' we interchange r and r' in (91). For the interior

flow, r' <a,

F(ka,

kr)IT)

cos n(OO')

(r <r' < a),

(93) I(1ca)

2ir

n

and for O < r' < r < a we interchange r and r' in (93).

Now by applying Green's theorem, (89) is converted to an integral equation

, ; s)

=

Jj Û(,

¿' ,'; s)g(', j';s) de' dy',

'íi(r, O; s) =

where g, as defined in (89), contains the unknown function i, and D is the domain of integration:

r' > a for the exterior flow and r' < a for

the interior flow. Upon substitution we obtain for the exterior flow,

r > a,

2ir D 2rv

J

{JFn(kr/ ka)

'fl

_{dr' +} K(ka) 7L- J O a

J F,(kr, ka) r)(z2_fFJ)(ûO±)r dr'}cosn(OO') dû , (95)

K(ka)

1

where F(x, y) is defined by (92). And for the interior flow, r < a,

(r, O; s) =

J {] F(kr',

ka)1

(f'J')(o+)r' dr'+

2rv z_.

_n-0

I(ka)

+

f

F(kr, ka) dr'lcos n(OO') dO'. (96)

j

J(1ca) j

O

Aside from the constant , the choice of which will affect the numerical evaluation of y, we note that (assuming y is integrable in D) the integrals in (95) and (96) all converge and the series also converges absolutely and uniformly for r > a or r < a and for O < O < 2ir. One practical method of solution for the above integral equation of y is by iteration. As the first iteration, we may approximate the in the integrand by setting O (which is consistent with boundary conditions of i), obtaining from (95) and (96) the first order solution i. The second iteration j3(2) is obtained by replacing j3( for jY in the integrand, and so forth. The term t0(r'; s) in the integrand is of course given by (53), or

t7'0(r; s) = (W/s)K0(kr)/K0(ka)} (r > a),

= (W/s){10(Jcr)/10(ka)} (r < a). (97)

The skin-friction r

and r_ on the outer and inner surfaces of the

cylinder has the following Laplace transform

(1 a(û0+)

5n _J5 a

By making use of the above results, and deriving &i/ar by differentiating the expression (95) or (96), together with use of the Wronskian

I(z)K(z)I(z)K(z) = liz,

422 T. C. WU AND T. YAO-TSU W[J we obtain

_i

xW X1(ka)

(f'Ira{5)

K0(ka) ,'

2,

Kn(kr')(2fJ)(+)Fd}

₍₉₉₎ K(1ca) n= _a ¿L cW 11(ka) !f'!raL/('i'9) 10(ka) 2,r a

--

_n-

f cos n(OO') dO' f

I(ka)

I(kr)

-f J')(ñ0+v)r'dr').

o o

Now we turn to the problem of choosing a. For large time, correspond-ing to small s and k, the factor K(kr)(0+) in (95) and (99) falls off at a much slower rate with increasing r, as r - , than in the case of small t, or large k. The rate of convergence of the integrand in (95) and (99)

at r =

can however be improved by the choice

= um (f'f'). (101a)

Furthermore, it may be pointed out that the transformation (82) may contain in f(fl an arbitrary scale factor of uniform magnification about

= 0. This scale factor can be sochosen as to render

= um l'i = 1.

(lOib)

This condition in turn determines the radius a of the circle in the -pIane. Under this condition the transformation (82) can be expressed in general

as

z =f()

+b1!+b2!2+...,

(102)

where b, are constant coefficients. Hence

=

1f'f' =

so that a2_f'f' is O(r2) as r =

- . By making use of the

asymptotic expansion of K(ka), K(kr), z

for small k = /(s/v), and

assuming i to be of the same order as (as can be verified a posteriori),

it can be shown from (99) that

J_ ¿4W Ki(aJ(sIv))0(1ogCoa Is

Jf'iks)Ko(a(s/v))

\ J

=

fr+eq(8;

=

(100) (103)

where (s; a) represents the value on the outer surface of the related circular cylinder of radius a (see (54)) and CO3 C are the coefficients defined in (58). We further note from (59) that

+eq(8;a) MW/as ){1+O(8log.S)}. (104)

log[C0a/(s/v)j

Thus the order term in (103) is lessimportant comparedwiththatin (104). Here, (103) exhibits a salient feature of the asymptotic behaviour of the skin-friction that for time large the local skin-friction of an arbitrary cylinder is asymptotically equal to that on the related circular cylinder multiplied by the scale factor ¡f'I- of the conformal transformation

evaluated at the surface while the radius of the circular cylinder is

determined by the condition Jf'J

i at the point of infinity.

This

result holds valid whether or not f' has any singularity (zero or infinity) on the body surface.

The total skin-friction drag of the arbitrary cylinder is given by

D

dl=

_{Ír+ f'! die,} (105)

where F denotes the contour around the original cylinder, and Fe, the contour around the related circular cylinder in the c-plane. For moderate values of t, D can be calculated from (105) by using the iterated result

of T in the integrand. The calculation is particularly simple for very large t, for in this case use of (103) yields the remarkably simple result

D

J T+eq die

=

Deq+O{T log(C0/4T)}2 (106a)

r,

as T

vt/a2

.-

_{c. Therefore the total drag experienced at large T by}

the arbitrary cylinder is equal to that acting on the related circular

cylinder; or by using (59), D n=0 Ca2 (fl+1)

+ou1T\Ì,

(_YAn(lo 4vt) (1 T (106b)

where the A are given by (59b) and C0 is defined in (58). As a few typical cases we mention the following examples.

(i) Elliptic cylinder. The region outside an elliptic cylinder of semi-major and semi-minor axes b1 and b2 is mapped into the out-side of a circle of radius a by

z

=f()

= +(BI),

(107a)

with

424 J. C. WTJ A2m T. YAO-TSU WU

the expression (107a) also satisfies condition (10 lb). This determines

the radius a of the related circular cylinder, and hence also the drag as given by (106b). The above result of an elliptical cylinder and the special case of flat strip given below are in agreement with that of Batchelor (4), who obtained this leading term by a somewhat simpler argument, and also in agreement with Hasimoto's result (7) which is obtained by using a conformal transformation method. (ii) Flat strip of chord c. This is a special limit of the above case when

b1 - c/2, b2

=

O so that

a

=

c/4

and B

c2/4 (108)

Hence the skin-friction drag of a flat plate of chord length c is

asymptotically equal at large time to that acting on a circular

cylinder with the diameter equal to half the chord length. The

counterpart of this flat-strip problem for the small time stage has been treated by H. Levine (15) using an integral equation method.

iii) Lenticular-arc cylinder.

Consider a symmetric lenticular

arc consisting of two circular arcs, of equal radius, subtending an inner vertex angle 2yr at the corners z

+c. The region outside the

lens is mapped into the outside of a circle of radius a in the -p1ane

by

_{(_a(1v)}

(109)

z+

\+a)

The condition that IdzIdl

-

i as Izi -+ cc gives

ly

(110)

9. Discussion on Rayleigh's analogy

Rayleigh (1) originally suggested that the relatively simple solution of the unsteady flow along an infinite flat plate could be used to give a

qualitative estimate of the steady plane flow along a semi-infinite flat plate, with the free stream velocity U perpendicular to the edge of the plate.

This method assumes that vorticity in the boundary layer

spreads laterally out into the stream at the rate given by the unsteady problem, but at the same time is convected downstream with the fluid. As a rough approximation, t may then be replaced by x/U, x being the streamwise distance measured from the leading edge of the plate. In consequence, since the actual convective velocity is less than U, the skin friction is overestimated and the boundary layer thickness is under-estimated. In fact, the local skin-friction coefficient calculated by this analogy is G

=

(2/ir)Re;, where Re

=

Ux/v;

this value is about

70 per cent in excess of the exact value Cf = 0664 Re; of Blasius.

Actually, the exact C.

is obtained if t is identified with X/Ue, U = 0346U being the effective convection velocity.

Similar suggestions have been made by various authors that the

solution of the present generalized Rayleigh's problem could be used, by the same analogy, to give an approximate solution of the more difficult non-linear problem of the steady flow along a semi-infinite body of the same cross-section. In extending the original Rayleigh's analogy, one

should of course realize that the corresponding steady problem is

characterized not only by the non-linear effect, but also by the effect of curvature of the solid boundary, and generally also by a non-uniform pressure field. For values of (vx/U), or the boundary layer thickness, small compared with a typical dimension i of the cylinder cross-section, then, by the analogy, the local skin-friction on the arbìtrary cylinder without corners is the same, to the first three leading terms (see (81)), as that on a circular cylinder of the same radius of curvature. When the cylinder has a number of corners, the total skin-friction could be corrected accordingly, after Batchelor (see (81a)). For large values of vx/ U, the boundary layer thickness is large compared with i, the effect of the cross-sectional shape becomes important, and it is in this range that we expect the Rayleigh analogy to give a more accurate result since the effective convection speed is closer to U. It would therefore be of interest to make comparisons for the cases when the exact solution of the steady flow is known.

As the first example, we consider the flow along a right-angled corner bounded by two planes y = 0, x > 0, and z = 0, z > 0, the flow being along the x-axis, for y > 0, z > 0. This steady flow problem was first treated by Carrier (16), but his result contains an error due to neglecting certain cross-flow terms. The corrections were later made by Carrier and Dowdall (17), the maximum correction calculated was shown by Dowdall (18) to be of the order of 4 per cent for the streamwise velocity. Carrier's solution of the x-component velocity was expressed in the form

u/U =f)f)+a2h(, )/

a,

(111) where = y(U/vx), = z(U/vx).

_f0()

is

the Blasius function,

satisfying the equation 2f" +ff" = 0 (0 < cc), and the conditions

f(0) =f'(0) = 0, f'(cc) = 1.

The second term, h,,, of (111) has been computed, with the cross-flow terms included, by Dowdall, using the relaxation method, showing that this term is of the order of 20 per cent of

f)f) at most. Now, direct application of Rayleigh's analogy to the

426 J. O. WtI AND T. YAO-TSU WIJ

unsteady solution (19), by replacing t by /U, and

Wu, by u, yields

u/U

=

erf(f2)erf(/2). (112)

Comparison with the true value (111) shows that the form of this result is correct, but that the value (112) is too large. This overestimate is easily seen to be of the same order as the difference between

f ()

and erf (i/2), which is the same as in the original Rayleigh's approximation.

However, if the U in the expressions for

and

is replaced by

U8

=

O346U, then the agreement is greatly improved, particularly for

small values of and . This further indicates that the approximate

solution of skin friction

=

t(u/az)2_0 should be reasonably accurate if U8 is used as suggested.

In the case of the corner flow along a wedge bounded by two planes

O

=

O, x > O, and O = o, x > O, the approximate solution of the

skin-friction by Rayleigh's analogy can be obtained simply by replacing t by x/U in (42); the behaviour of

r

as the radial distance r

-

O is the same as was stated following (42). This approximate solution has also been discussed by Sowerby and Cooke (19).

We next consider the steady flow along the outside of a semi-infinite circular cylinder of radius a. The method of expansion in series can be used to evaluate a solution that is valid near the leading edge. This solution was obtained by Seban and Bond (20); Kelly (21) later gave some important numerical corrections to the result of Seban and Bond. Far downstream, where the boundary layer is thick compared with the cylinder radius, a solution may be obtained by use of an asymptotic expansion. This was done by Stewartson (22). There is an extensive region in which neither of the series representations is valid; a Pohl-hausen approximate method, however, can be used here. Glauert and Lighthill (23) used a combination of these methods and a careful choice of velocity profile to derive results which appear to be reliable over the entire length of the boundary layer. These results for the skin-friction

coefficient, C1

=

T/pU2, are recited in the following equations.

C1(KSB)

O664(Rey+ F392 () Re1

1.594()Re;+...

(113)

+---_(2 log 2+2)G3+O(G4) ,

(114)

ReaG

where G

=

log (4vx/C0Ua2), C0

=

e7, y

=

O5772.... Here, K-8-B stands

for Kelly-Seban-Bond, and G-L-8 for Glauert-Lighthill-Stewartson. The corresponding coefficients in the solution of Glauert-Lighthill, for

Rayleigh's analogy to (56) and (59a) yields

Cf '-' 113 Re; +Res_O.283()Re

+O(()Re;2}

(115)

C,

{G'(*r'y)03+O(04)}

(116)

Rea

Comparison between (113) and (115) shows that the approximate

solution overestimates Cf by a factor about the same as in the case of a flat plate. Although the leading term could be improved by using Ue instead of U, this same Ue does not improve the higher-order terms. For large values of vx/ Ua', it is noted that the coefficients of the first two terms, G-' and 0', in (114) and (116) are in complete agreement. This accuracy of the approximate solution may be attributed to the fact that the flow in a large part of the outer boundary layer approximates very closely to the free stream flow, and that the fluid acceleration becomes negligibly small in this range.

REFERENCES

LoRD RAYLEIaH, Phil. Mag. 21 (1911) 697.

L. HOWARTH, Proc. Camb. Phil. Soc. (Math. phys. Sci.) 46 (1950) 127. L. SOWERBY, Phil. Mag. 42 (1954) 176.

G. K. BATCHELOR, Q. Ji. Mech. appi. Math. 7 (1954) 129. J. C. Coox, J. phys. Soc. Japan, 11 (1956) 1181.

Q. Ji. Mech. appl. Math. 10 (1957) 312. H. HAsIM0TO, J. phy. Soc. Japan, 6 (1951) 400.

J. phys. Soc. Japan, 9 (1954) 611. J. phys. Soc. Japan, 10 (1955) 397.

H. S. Cxsiw arid J. C. JAEGER, Conduction of Heat in Solids (2nd edn., Oxford University Press, 1959).

G. N. WATSON, A Treatise on the Theory of Bessel Functions (2nd edn., Cambridge

University Press, 1948).

A. ERDLYI et al., Higher Transcendental Functions, Bateman Manuscript Project (McQraw-Hill, 1953).

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D. D. MALLICK, Z. angew. Math. Mech. 37 (1957) 385. H. LEvXHE, J. Fluid Mech. 3 (1957) 145.

G. F. CARRIER, Q. appi. Math. 4 (1946) 367-370.

G. F. CARRIER and R. B. DOWDALL, Note on the solution of boundary layer in a corner (unpublished).

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H. R. KEtiY, J. aeronaut. Sci. 21 (1954) 634. K. STEWAIeTSON, Q. appl. Math., 13 (1955) 113.

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